Mean Free Path of Gas Molecules
Calculate the mean free path and collision frequency of gas molecules from temperature, pressure, and molecular diameter.
Based on kinetic molecular theory.
The mean free path (λ) is the average distance a gas molecule travels between successive collisions.
Formula:
λ = kB × T / (√2 × π × d² × P)
Where:
- kB = 1.381 × 10⁻²³ J/K (Boltzmann constant)
- T = temperature (Kelvin)
- d = molecular diameter (meters)
- P = pressure (Pascals)
Collision frequency (z) — collisions per second:
z = v_avg / λ
where v_avg = √(8kBT / πm) = average molecular speed, and m = molecular mass in kg.
Approximate mean free paths at 25°C, 1 atm:
| Gas | d (pm) | λ (nm) |
|---|---|---|
| N₂ | 370 | 67 |
| O₂ | 346 | 76 |
| H₂ | 289 | 110 |
| He | 258 | 137 |
| CO₂ | 450 | 45 |
Effect of pressure: λ ∝ 1/P — halving pressure doubles the mean free path. At 0.1 mPa (high vacuum): λ ≈ 68 m for N₂, so molecules cross the chamber and hit a wall long before they meet each other. This is why vacuum systems are described by whether the mean free path exceeds the chamber size, not by pressure alone.
Knudsen number: Kn = λ / L (where L is characteristic system length)
- Kn « 1: continuum flow (many collisions, classical fluid dynamics applies)
- Kn » 1: molecular flow (molecules collide with walls more than each other)
- Kn ≈ 1: transitional regime
Applications:
- Vacuum technology (mean free path determines vacuum quality)
- Semiconductor manufacturing (deposition processes require Kn > 1)
- Atmospheric science (transport coefficients)
- Thermal conductivity and viscosity of gases
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